Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Round to the nearest hundredth. $$122.62^{\circ}$$

Step-by-Step Solution

Answer

122.62° is approximately 2.14 radians.

Step 1: Understanding the Relationship

Degrees and radians are two units for measuring angles. To convert degrees to radians, you can use the conversion factor that relates the two units: \( \pi \, \text{radians} = 180^{\circ} \). This means \( 1^{\circ} = \frac{\pi}{180} \, \text{radians} \).

Step 2: Setting Up the Conversion

We need to convert \( 122.62^{\circ} \) to radians. Use the conversion factor from Step 1: \( 122.62^{\circ} = 122.62 \times \frac{\pi}{180} \, \text{radians} \).

Step 3: Performing the Calculation

To perform the conversion calculation, first multiply 122.62 by the factor \( \frac{\pi}{180} \).\[\text{Radians} = 122.62 \times \frac{\pi}{180} \approx 122.62 \times 0.0174533 \]

Step 4: Final Calculation and Rounding

Calculate \( 122.62 \times 0.0174533 \) to find the radians measure.\[\text{Radians} \approx 2.140465826 \]Now, round this result to the nearest hundredth:\[\text{Radians} \approx 2.14 \].

Key Concepts

Degrees to Radians

Converting degrees to radians involves a straightforward mathematical process. The two units, degrees and radians, are different ways to measure angles. While degrees use a base of 360 for a full circle, radians use a base of 2\(\pi\). This difference is like comparing inches to centimeters in length. The conversion factor between degrees and radians is based on the equivalence of a full circle:

• One full circle in degrees is 360°
• One full circle in radians is \(2\pi\)

Therefore, we can derive that:\[ \pi \, \text{radians} = 180^{\circ} \] so the conversion of degrees to radians can be done using the factor: \[ 1^{\circ} = \frac{\pi}{180} \, \text{radians} \] To convert an angle, simply multiply the degree measure by \( \frac{\pi}{180} \) to get the equivalent in radians.

Angle Measurement

Angle measurement is crucial in many fields such as engineering, architecture, and trigonometry. It allows us to precisely describe the orientation between two lines starting from a common point, known as the vertex. To measure angles, we can use different units such as degrees and radians, each suitable for different purposes. Here’s why these units matter:

• Degrees: Often used in day-to-day applications like navigation or weather conditions. The circle is divided into 360 equal parts or degrees.
• Radians: More prevalent in higher mathematics, especially in trigonometry and in functions involving calculus. Radians relate more naturally to other mathematical concepts.

Understanding both units enhances your ability to apply them in different mathematical contexts and conversions is key to flexibility in solving problems.

Unit Conversion

Unit conversion is a mathematical process that lets you change one type of measurement unit into another, while maintaining the precise size or quantity being measured. This skill is especially important across scientific disciplines.When converting angle units from degrees to radians or vice versa, it involves using a formula. Here it follows these steps:

• Determine the conversion factor, like \( \frac{\pi}{180} \) from degrees to radians.
• Multiply the original measurement by the conversion factor.
• Perform the arithmetic to find the new value in the desired unit.

This ability not only ensures accuracy but also opens up a better understanding of how the two systems relate, allowing for greater versatility in mathematical problem-solving.

Trigonometry

Trigonometry is a branch of mathematics that focuses on angles, sides, and the relationships between them in triangles. It relies heavily on angle measurements in both degrees and radians. Some trigonometric functions like sine, cosine, and tangent use these units, making it vital to understand unit conversions between degrees and radians. When angles are measured in degrees or radians within trigonometric problems:

• Radians offer a natural way to express angles as they directly relate to the circumference of a circle.
• Knowing how to switch between these units helps solve problems more flexibly.

Dive deeper into trigonometry, you'll often find radians more efficient to handle periodic functions and solve calculus-related problems. Understanding both units and conversions will enhance your ability to tackle a wide range of mathematical problems. The harmony of angles, sides, and their calculations found in trigonometry can be vastly broadened by mastering these conversions.